VIBRATION INSTITUTE 33 ANNUAL MEETING, HARRISBURG, PA. , JUNE 23-27, 2009R D

UNDERSTANDING AMPLITUDE AND PHASE IN ROTATING MACHINERY

Edgar J. Gunter, Ph.D.

Professor Emeritus

Department of Mechanical and Aerospace Engineering

Former Director, Rotor Dynamics Laboratory

University of Virginia

RODYN Vibration Analysis, Inc.

DrGunter@aol.com

Abstract:The measurement of phase is essential in the balancing of single- or multi-plane balancing of rotating machinery. The

rate of change of phase is important, as it may indicate the presence of a critical speed, and from the rate of change of

phase one may be able to deduce the amplification factor or log decrement of a particular mode. Having a phase

reference mark is particularly critical on complex machinery such as gas turbines and also the space shuttle oxygen and

hydrogen pumps. Without the presence of a phase signal, it is impossible to determine what component of the total

vibration is synchronous and what is due to external or subsynchronous vibrations. The use of the timing mark, therefore,

allows us to track amplitude and phase to determine critical speeds, amplification factors, and balancing.

Keywords: Phase; Bode Plots; Nyquist Plots; Amplification Factor; Critical Speeds; Log Decrement

Introduction:

The understanding of amplitude-phase relationships is extremely important for all users of turbo-machinery, as well as

the manufacturers. One of the greatest problems in the diagnosis and correction of field problems with rotating

machinery is the lack of adequate vibration instrumentation on the machinery to observe the vibrations, the lack of

adequate balancing planes, and most important, the lack of a phase reference signal. The field installation of vibration

instrumentation and a phase indication probe are often very difficult and time consuming to achieve. A great deal of

time, effort, and efficiency would be achieved if manufacturers had adequately designed the machinery with the idea of

sufficient instrumentation, balance planes, and phase reference signal for machine diagnosis and balancing.

For example, without an adequate reference signal it is difficult to separate synchronous from nonsynchronous motion

on a turbo-rotor and makes the problem of field balancing extremely difficult. Some notable examples that will be

discussed are the correction of the failure problem encountered with the space shuttle main engine oxygen pump. The

oxygen pump vibration failure is an example in the extreme of what happens without the application of a suitable

reference signal. The vibration of interest in the oxygen pump was initially drowned out by the high intensity, high

frequency vibrations generated by the rocket engine. It was only after a reference signal was generated laboriously from

the four-lobed speed pickup that the vibration could be suitably analyzed and the pump design corrected. Once the

timing mark was secure, synchronous tracking filters showed that the oxygen pump was operating at a second critical

speed. The pump seals then could be redesigned to suppress this critical speed in order for the pump to operate safely

through the entire speed range. Another notable example was the attempt to analyze the vibration characteristics of a

typical commercial small jet aircraft engine. Although years of vibration data had been taken on this engine, it proved

to be worthless since there was no corresponding phase reference mark to determine the nature of the critical speeds and

the amount of damping in the modes. New data had to be generated, which included a timing reference mark, in order

to ascertain the problems. Many field problems in vibration diagnostics and field balancing could be more easily

resolved if adequate provisions had been initially made to provide adequate phase measurements.

1

What is Phase and What Does it Mean?:

The electrical engineers are well acquainted with phase The phase of two or three voltages such as 3 phase motors is

well established and the understanding of phase and power factor is critical in electrical equipment. The roll of phase

and its measurements in rotating machinery is not as well established or appreciated by many manufactures.

Phase Measurements With Strobe Light:

One of the earliest methods of phase measurements was the use of a strobe light which was triggered by a velocity

pickup. Fig. 1 shows the phase position before and

after the placement of a trial weight on a rotor

balancing demonstrator.

The phase measurements are made from the stationary

compass. The initial strobe light shows the arrow

pointing at 270 deg. After placement of the trial

weight, the strobe light shows that the phase has shifted

60 deg in the clockwise direction.

Although balancing may be made from such

measurements, the strobe light procedure of relative

phase measurements provides little insight to the actual

rotor dynamics amplitude-phase relationship.

The strobe light is triggered to flash based on the signal received from the analyzer. The strobe light is made to flash

on the shaft, freezing it in place. When a strobe light is used for a phase reference in balancing, the determination of the

rotor high spot is not as simple or straightforward as when a noncontact probe and a keyphasor reference mark are used.

Fig. 2 , developed by C. Jackson, represents the phase

angle for a typical seismic vibration analyzer in the

displacement mode. In addition to the 90 deg phase

shift due to integration of the velocity signal, there is

a speed-dependent rolloff phase shift due to the mass

inertia effects of the velocity sensor at low frequencies.

Fig 2 was developed by Jackson to help find the rotor

heavy spot for assistance in the placement of the initial

trial weight for influence coefficient balancing.

Amplitude and Phase Measurements With

Noncontact Inductance Probes:

A noncontact probe has the distinct advantage of

observing shaft motion directly. One of the earliest

types of noncontact displacement probe dating to the

1960's was the British Wayne-Kerr capacitance

probe..This probe was ideally suited for early gas

bearing rotor dynamic studies but was not practical for

industrial rotating equipment. This was due to the cost of the equipment, problems of contamination of the probe surface

and limited range of motion.

Fig. 1 Phase Readings With 2 gm Trial Weight

Fig 2 Phase Lag From Seismic Velocity Sensor to Rotor

Heavy Spot ( Jackson, 1979)

Fig. 3 Shaft Motion With Noncontact Displacement Probe

2

Fig. 3 represents the direct shaft displacement as observed by a noncontact probe. To view this motion, the vibration

signal was displayed on an oscilloscope and the motion recorded by a Polaroid camera. No phase measurements were

recorded with this early procedure. By taking a series of pictures at various speeds, a crude plot of amplitude vs speed

could be attained.

A major advancement in the understanding of rotor dynamic amplitude-phase relationships was the introduction of the

Bently noncontact inductance vibration pickup. Unlike the Wayne Kerr capacitance probe, the inductance probe was

much less expensive, rugged, and with a larger range of motion. A unique innovation of Donald Bently, before the

advent of the FFT analyzer, was the introduction of the keyphasor

Fig. 4 represents a typical setup with an inductance probe to monitor shaft motion and the addition of a second probe

referred to as the keyphasor probe. A grove is placed on the shaft in line with the keyphasor probe. When the shaft

notch passes under the keyphasor, a signal is generated as shown in Fig. 5. When the output of the keyphasor is fed into

the z-axis of an oscilloscope, a bright spot occurs on the vibration signature as shown in the lower waveform as seen in

Fig. 5.

The phase convention, in this case is the angle phi, as measured from the bright spot on the wave form of Fig 5 to the

peak amplitude. This angle has a practical significance in balancing. For example, if one lines up the timing reference

mark with the keyphasor probe, the peak amplitude A occurs at the angle phi as measured from the inductance probe

in the direction opposite to rotation. This point is often referred to as the “high spot” The location of the unbalance is

referred to as the “heavy spot”.

If we know the approximate critical speed of the rotor, then we have an idea of where to place the first trial weight for

field balancing for a relatively simple rotor system. If the rotor is operating in the subcritical speed range (well below

the first critical speed), then the high spot and the heavy spot are in the same vicinity. The first initial trial weight would

be placed 180 deg from the high spot. If the rotor were operating well above the first critical speed, for the case of the

simple single mass Jeffcott rotor, the mass center is out of phase to the maximum displacement or high spot. In this case

the initial trial weight would be placed at the angle phi as measured counter rotation from the inductance probe. This

placement seems against intuition. If we should attempt the unusual balancing procedure of balancing at the critical

speed, then in this case the mass center is leading the mass center by 90 deg. In this case one would line up the keyphasor

probe with the notch and place the first trial weight at an angle of 90 +phi deg as measured opposite of rotation from the

inductance probe. Such a direct understanding of the relationship between the peak amplitude and the approximate

location of the unbalance is not possible with the strobe light method.

Amplitude and Phase Relationships For the Jeffcott Rotor:

The first English publication to describe the dynamics of the flexible single mass rotor was presented by H.H Jeffcott

in 1919. He correctly expressed the equations of motion of the single mass rotor with unbalance and damping and

presented an explanation of how the rotor could pass successfully through the critical speed. He also showed the

displacement-phase relationship at speeds below, at and above the critical speed.. To properly understand rotor dynamics

amplitude-phase relationships and their implications, we must have a good understanding of the simple Jeffcott rotor.

Fig. 5 Amplitude Signal With Keyphasor MarkFig. 4 Noncontact and Keyphasor Probes

3

One of the earliest accounts on rotor dynamics was the article presented in 1869 by W. A Rankine entitled “On the

Centrifugal Force of Rotating Shafts”, Engineer, Vol 27,London. Rankine introduced the elementary concept of

indifferent rotor equilibrium. Rankine examined the equilibrium conditions of a frictionless, uniform shaft disturbed

from its initial position. Because he neglected the influence of the Coriolis force, he concluded that: motion is stable

below the first critical speed, is neutral or in “indifferent “ equilibrium at the critical speed , and is unstable above the

critical speed.

During the next half century, this analysis lead engineers to believe that operation above the first critical speed was

not possible. It was not until 1895 that DeLaval demonstrated experimentally that a steam turbine was capable of

sustained operation above the first critical speed.

Fig. 6 represents the single mass Jeffcott rotor. In this

model the mass is concentrated at the shaft center and the

disk is considered as massless. The bearings at taken as

simple supports. The damping is applied at the disc center.

This assumption may also be considered as a form of

model damping. The unbalance in the disk is created by

uthe small offset e of the disk center from the axis of

rotation.

The equations of motion of the single-mass Jeffcott rotor

may be combined into one complex vector equation by

means of the complex variable transformation Z = X + I

Y as follows:

The steady state synchronous unbalance response due to unbalance is given by:

(4)

The amplitude and phase may be expressed in dimensionless form by using the following variables:

= natural frequency on rigid supports ; = frequency ratio

= critical damping ; = d am p ing ra t io ; A = = dim amplitude

The relative phase angle lag â of the deflection vector from the rotating unbalance load is given by

(5)

Fig 6 Single Mass Jeffcott Rotor

4

Amplitude and Phase of the Jeffcott Rotor Below, At and Above The Critical Speed:

There are 3 distinct ranges of motion for the Jeffcott rot In each of these ranges, the phase angles are distinctly

different. The first speed range of interest is the speed range referred to as the subcritical speed range.

In this speed range we have the condition that the operating speed is much lower than the critical speed.

Case 1 -Subcritical Speed Operation

Thus we have the condition that f<< 1 and the dimensionless amplitude and phase angle is approximately given by:

A = f (6) 2

â = 0 (7)0

This implies that at low speeds well below the first critical speed, the amplitude of motion due to unbalance excitation

is increasing as speed squared. This region is characterized as rigid body motion region. Many current industrial

fans are designed to operate in this region as rigid body rotors often with highly undesirables effects. The location

of the mass unbalance is in phase with the maximum deflection. In this low speed region, we say that the high spot

and the heavy spot coincide

Case 2 - Critical Speed Operation

The second region is a unique region in which the rotor is operating exactly at the critical speed.

In this case f=1 and the dimensionless amplitude of motion is given by:

(8)

The mass center is leading the maximum deflection by:

â = 90 (9)0

Where Acr is referred to as the critical speed amplification factor on rigid supports. This is a finite value depending

upon the modal damping acting upon the rotor. It should be noted that with the Jeffcott rotor all of the damping is

concentrated at the disk. The bearings are providing no damping since they are node points.

With a rotor supported on rolling element bearings, the amount of aerodynamic drag damping available at the center

disk can be quite small. This could lead to dangerously high levels of vibration if such a rotor were to be

continuously under these conditions.

Case 3 - Supercritical Speed Operation

In the supercritical speed the rotor speed is well above the critical speed. This region is referred to as supercritical

speed operation in which the dimensionless speed f >> 1. The dimensionless amplitude of motion is given by:

A = 1 (10)

In this region of operation, the amplitude of motion is no longer increasing as mass center inversion has occurred.

The rotor is spinning or rotating about the center of mass. Since the center of mass is offset from the axis of rotation,

uthe inductance probes observe an orbit with radius e ...

The phase angle of motion â is now out of phase to the displacement vector and is given by:

â = 180 (11)0

Fig. 7 shows the motion of the Jeffcott rotor in the speed ranges.

In the first figure, the speed is well below the critical speed. Point M represents the location of the mass center. Point

C is the center of rotation of the shaft. The point H represents the high spot and is in line with the mass center, line

C-M extended. At this low speed, the balance correction weight would be placed opposite the high spot. The center

figure represents the rotor operating at the critical speed. In this case, the rotor mass center is leading the maximum

amplitude or high spot H by 90 deg. If one were to mark the high spot, ( as I had once observed in industry on an

experimental rotor with caulk), one would place the balance correction at 90deg from the mark opposite the direction

of rotation. The third figure on the right represents the shaft motion well above the critical speed. The mass center

has inverted 180 deg and is now opposite the high spot .

5

Development Of A Jeffcott Rotor Using DyRoBes Rotor Dynamics Software:

A single mass rotor model was developed using DyRoBes to simulate an idealized Jeffcott rotor. A 200 lb

disc was mounted on a steel shaft of 40 inches length with a diameter of 4 inches. To simulate rigid

bearings, bearing stiffness values of 1.0e7 Lb/in were assumed. Since the shaft weight is included in this

model, the modal weight is the sum of the disk weight Wd plus one-half the shaft weight Ws. The total

Jeffcott modal weight then is approximately 271 lb.

Fig. 8 represents the Jeffcott rotor on stiff bearings mounted on a 40 inch shaft.

Fig. 9 represents the animated 1 critical speed of the Jeffcott rotor..It should be noted that with the Jeffcottst

rotor, there is no bearing motion. Anytime this situation occurs in a real rotor one has a dangerous

condition in which the bearings will provide no damping. In order to provide damping for the Jeffcott rotor,

a third “bearing” was applied acting at the disc center. At this bearing location, damping in both x-y

directions of 20 lb-sec/in was assumed.

The critical damping for the Jeffcott rotor is given by:

cr Cc = 2 M ù = 2 271lb/386.4x628 rad/sec = 881 lb-sec/in

For an assumed amplification factor of 20, î = 1/40 =0.025

The amount of damping acting at the disk center is given by C =0.025 Cc = 22 lb-sec/in

Fig. 7 Jeffcott Rotor Motion Below, At and Above The Critical Speed

Fig. 8 Jeffcott Rotor on Stiff Bearings Fig. 9 1 Critical Speed of Jeffcott Rotor at st

6,000 RPM

6

Fig. 10 represents the damped mode shape of the Jeffcott

rotor with damping of 20 lb-sec/in added at the disk

center. The addition of the damping causes a slight

reduction of the damped natural frequency to 5998 RPM.

The modal damping is expressed in terms of the log

decrement ä.

Therelationship between log decrement, æ; the damping

ratio î and critical speed amplification factor is as

follows:

(11)

(12)

Therefore, for low values of the damping ratio î, the critical speed amplification factor Acr is given in terms

of log decrement as ð /æ. The log decrement for the 1 forward rotor mode as show in Fig 10 is Acr = ð/.144st

= 21.8. This high amplification factor could be expected with a rotor operating on rolling element bearings.

Synchronous Unbalance Response and Phase With Jeffcott Rotor:

Figure 11 represents the unbalance response and phase of motion of the Jeffcott rotor with 2 oz-in of unbalance.

The motion is shown for the x probe. The unbalance is assumed to lined up with the keyphasor probe. Here

we see the classical 180 deg phase shift with the Jeffcott rotor. The motion plotted is for the center span of the

rotor..If a probe were situated at the center span of a multi-mass rotor or large turbine, we would see a similar

type of phase shift. Figure 11 represents the typical response observed with the Jeffcott rotor. At speeds below

the critical speed, the amplitude increases with speed squared. There is very little shift in phase.

Fig. 10 Damped 1 Mode of Jeffcott Rotor With st

20 Lb-Sec/In Damping Applied At Disc

Fig. 11 Synchronous Unbalance Response and Phase of Jeffcott Rotor

7

As we approach the critical speed, we see a rapid change in the phase angle. The rate of change of phase is

dependent upon the amount of damping present in the rotor system. As we approach speeds well above the

critical speed, there is no further change in phase. This is because the mass center has inverted. The rotor spin

axis now passes through the rotor mass center. The displacement is now 180 deg out of phase to the mass.

Determination of Amplification Factor From Amplitude and Phase Response:

There are several methods in which the rotor amplification factor may be determined

One of the most common methods taken from electrical engineering is the half power point method. At the

critical speed, the speed increment ÄN is measured at a height of 0.707 Xmax at the critical speed. In this case

the maximum amplitude at the critical speed is seen from Fig. 11 to be 21.665 mils. Therefore at the height of

0.707x21.666 of 15.3 mils, the band width is measured to be 286 RPM.

The critical speed amplification is given by:

(13)

This number agrees well with the computed amplification factor based on the calculation of the log decrement

from the damped whirl mode as shown in Fig.10.

A second method used for the determination of the system amplification factor is to determine the rate of change

of the phase angle at the critical speed. For example, the greater the measured rate of change of phase while

passing through the critical speed, the higher the amplification factor. Mathematically, this is determined by

differentiating the rate of the phase angle â (or ö ) with respect to the dimensionless frequency ratio f. By

evaluating this rate of change at the critical speed, given by f=1, we obtain the following:

(14)

Ths method is often employed with FFT analyzers for the determination of amplification factors.

A third method for determination of the amplification factor from the unbalance response curve is to divide

the peak amplitude at the critical speed by the amplitude at speeds well above the critical speed. A high speeds,

uthe rotor amplitude of motion is equal to unbalance eccentricity e . Dividing the maximum amplitude by the

high speed amplitude yields the critical speed amplification factor as follows:

(15)

Jeffcott Rotor 3 Dimensional Modes at Various Speeds:

Fig. 13 Jeffcott Rotor Motion at the Critical

Speed - 6,000 RPMFig. 12 Jeffcott Rotor Motion at 4,000 RPM

8

Figure 12 represents the 3 dimensional rotor motion at

4,000 RPM which is a speed of two-thirds of the critical

speed. It is seen that the unbalance vector and amplitude

of motion are essentially in phase.

Figure 13 shows the rotor motion at the critical speed of

6,000 RPM. In this case, the rotating unbalance vector is

leading the displacement vector by 90 deg. For slight

speed changes either above or below this speed, the phase

angles will rapidly change. This makes balancing near the

critical speed most difficult.

Figure 14 shows the Jeffcott rotor motion at a speed of

2,000 RPM above the critical speed. Since the rotor

system is lightly damped, this increase in speed is

sufficient to cause an additional 90 deg shift in phase. We

now have a situation in which any further increase in

speed will not result in any further increase in either

amplitude or phase. The mass center is now fully

inverted.

Polar Plot of Amplitude And Phase

A useful procedure for viewing the rotor motion is to

make a polar plot of the amplitude and phase of the

motion. Figure 15 represents the polar plot for the

Jwffcott horizontal motion. The plot of the phase angle

moves opposite of the direction of rotation. By viewing

the polar plot, one may obtain a reasonable ideal of the

direction in which to place the correction weight and also

the magnitude of the correction weight. For low values

of damping, the polar plot is almost a perfect circle. A

tangent to the start position and final position shows the

plane of the unbalance. The initial trial weight would be

placed at 180 deg from the 0 position. In this plot, the

peak to peak motion is plotted.

The distance between the initial 0 position and final

uposition represents 2 x e , the unbalance eccentricity

uvector. The value of e in this case is approximately 0.5

mils. The balance correction is of the order of the unbalance eccentricity times the modal weight is computed

to be approximately 2.2 oz-in. At a balance radius of 12 inches this is about 5 grams of correction. For an

initial trial weight a lesser amount of several grams would be appropriate.

One very important consideration of the operation of

a Jeffcott rotor at or near the critical speed is the

bearing forces transmitted. At all speeds below the

critical speed, the bearing forces transmitted at

greater than the rotating load. At the critical speed of

6,000 RPM, the unbalance of 2 oz-in creates a

rotating load of 127 lb or about 64 lb/bearing. The

actual transmitted bearing force is almost 1600 lb

This is a dynamic transmissibility of 25!

For operation at speeds exceeding the critical speed,

bearing forces are greatly reduced.

Fig. 14 Jeffcott Rotor Motion Above the Critical

Speed - N = 8,000 RPM

Fig. 15 Polar Plot of Horizontal Motion

Fig. 16 Bearing Forces Transmitted

9

Amplitude and Phase of Jeffcott Rotor With Shaft Bow and Unbalance:

The influence of shaft bow may have some unusual effects on the rotor response and also the phase of motion.

The effect of shaft bow produced a forced response on the rotor which is constant in magnitude as compared

to the effect of unbalance which is a speed squared effect. At the critical speed, the influence coefficients for

both effects are identical. Hence at lower speeds, shaft bow has a greater effect, while at speeds above the

critical speed, the influence of rotor unbalance predominates. A most interesting effect occurs when the shaft

bow is out of phase to the unbalance vector.

In Fig. 17, the shaft bow vector is exactly out of phase with the unbalance vector. The bow vector is .5 mils

at 180 deg. At low speeds, the amplitude is 1 mil p-p and is at 180 deg as shown in the upper phase plot. At

low speeds, the 2 oz-in unbalance has very little influence on the rotor response. As speed increases, the

influence of unbalance grows as speed squared until the rotating unbalance force exactly matches the force

generated by the shaft bow. When this occurs, the rotor amplitude goes to zero and the two force systems are

balanced. For this example, the balancing speed is around 6250 RPM. As this speed is increased above this

balancing speed, the amplitude begins to increase slightly until the rotor is operating or spinning about is mass

center. Since the amplitude plot as shown in Fig. 17 is peak to peak motion, the limit of motion for the Jeffcott

urotor is 2 x e . The 2 oz-in causes an unbalance eccentricity of 0.45 mils. Therefore, the maximum p-p high

speed runout to be observed will be 0.9 mils as shown in Fig. 17.

To better understand the interaction of shaft bow and unbalance, we need to examine the polar plot of Fig.18.

It should be noted, that with a lightly damped system, the polar plot forms very close to a circle. The origin of

the circle starts off at 1 mil at the phase angle of 180 deg. The phase change is in the counterclockwise

direction opposite to the direction of rotation. From this plot, it is seen that the polar orbit passes through the

origin of the coordinate system at 6200 RPM.

General Relationship Between Various Phase

Angles And the Keyphasor Probe:

Figure 19 shows the general cross section of the

Jeffcott rotor with the arbitrary placements of the

unbalance vector and the keyphasor probe. The

noncontact probes observe the rotating vector ä

which is represented by line O-C. The mass center

M is displaced from the axis of rotation by the small

ueccentricity vector e . The angle between the mass

center vector and displacement vector is the angle â.

The angle actually observed and displayed in the

Bode plot will be the angle ö since in general, in a

real rotor, the mass center will not be aligned with

the keyphasor mark.

Fig. 18 Polar Plot With Shaft Bow Out of Phase to

Rotor Unbalance

Fig. 17 Bode Plot With Shaft Bow Out of Phase

to Rotor Unbalance

Fig. 19 Deflected Rotor With Keyphasor Probe

Showing Mass-Displacement Phase Relationships

10

Jeffcott Rotor With Coupling Weight:

Figure 20 represents the Jeffcott rotor with a shaft extension a 20 lb coupling weight attached. The addition

of the coupling weight causes a reduction of the critical speed from 6,000 RPM to 5,708 RPM as shown in

Fig.21. In Fig. 21, it is seen that the coupling motion is out of phase to the center span disc motion. At higher

speeds, a second critical speed at 16,890 RPM will be encountered. This mode is mostly due to the dynamical

behavior of the coupling with little motion at the central disk. Large coupling overhangs are to be avoided due

to their highly undesirable behavior at elevated speeds.

The coupling location is often used for a trim balancing plane often with unusual phase change results.

Figure 22 shows the coupling end motion with a trim balance weight of 2 oz-in added in-phase to the 2 oz-in

unbalance on the center disc. The critical speed is at 5,700 RPM as predicted by the critical speed analysis.

The amplification factor observed at the coupling location is approximately 23. As the rotor passes through

the critical speed region, there is a phase inversion between the mass unbalance on the disc and the coupling.

This mass inversion leads to an apparent local balancing of the coupling at around 7,000 RPM. The phase angle

initially has gone through a 180 deg phase change, which is to be expected for the Jeffcott rotor. The reversal

of phase then occurs. After a 90 deg reversal has occurred, we obtain a nodal point with zero motion at the

coupling.

This point is sometimes referred to as the “anti resonant” speed and corresponds to a 90 deg phase shift reversal.

A similar effect will also be observed with tuned support systems. As the speed is increased above the anti-

resonance speed, the coupling amplitude continues to increase. Note that the coupling motion is continuing to

increase with no apparent increase or change in the observed phase angle. The reason for this is because the

next critical speed with high coupling motion is not encountered until we reach the 16,000 RPM speed range.

Fig. 21 1 Mode Jeffcott Rotor With ST

Coupling Weight - Nc1 = 5,708 RPM Fig 20 Jeffcott Rotor With Coupling Weight

Fig. 23 Coupling Response With 4 Oz-In Ub

Out of Phase to Disc Unbalance

Fig. 22 Coupling Response With 2 Oz-In Ub

In Phase With Disc Unbalance

11

Figure 23 shows the coupling motion with 4 oz-in unbalance out of phase to the disk unbalance. The phase

behavior for the coupling end is most unusual as there are two observed anti-resonance speeds created. The

first is at low speed around 4,000 RPM. Although the amplitude is small, there is a 180 deg phase shift

associated with with speed. The second anti-resonance speed is the higher self balancing speed around 8,500

RPM. There also is a corresponding 180 deg phase reversal associated with this speed. Thus it is apparent that

when we observe unusual phase shifts and reversals in phase in apparently simple systems, then it might be

worthwhile to examine the coupling and its balance condition.

Figure 24 shows the motion near the disk center with the large coupling applied out of phase to the disk

unbalance. The behavior appears to be very similar to the expected behavior for the Jeffcott rotor except for

the behavior around 4000 RPM. Since the coupling unbalance is out of phase to the disk unbalance, we

encounter a balancing or anti-resonance condition at this speed. There is an associated reverse phase shift that

occurs corresponding to the nul vibration speed. Although the amplitudes of motion in this low speed range

are small, the phase change behavior should not be ignored as this may indicate some external effects to the

rotor such as the coupling. Fig. 25 shows that coupling unbalance may lead to high bearing forces at speed.

Fig. 24 Motion Near Center Rotor Span With Large 4 Oz-In

Coupling Unbalance Out-Of-Phase to the Disk Unbalance

Fig. 25 Coupling End Bearing Forces Transmitted With Large

Out-Of-Phase Coupling Unbalance 12

Amplitude and Phase Behavior For Multimass Rotors:

Figure 26 represents a multimass rotor with the addition of two more disks. Figure 27 represents the first critical

speed and mode shape. In general, with multistage compressors and turbines, the first mode shape will be

similar in form to the Jeffcott rotor.

The addition of the additional disks causes a second critical

speed to be within the operating speed range of the rotor.

Figure 28 represents a critical speed map to include the first

two critical speeds as a function of bearing stiffness. It can

be seen that for an operating speed range of 10,000 RPM,

a second critical speed may easily fall within the operating

speed for bearing stiffness values that fall below 300,000

Lb/In.

To avoid this apparent problem, compressor manufacturers

would often use highly preloaded tilting pad bearings with

bearing stiffnesses exceeding 1.0E6 Lb/In. Although the

second critical speed is well removed from the operating

speed, this practice would result in very serious rotor

aerodynamic instability problems due to reduced modal

damping caused by the excessive bearing stiffness.

Tilting Pad Bearing Coefficients:

Figure 29 represents a 5 pad bearing with load on pad. The bearing stiffness is asymmetric in the horizontal and

vertical directions. Hence the synchronous amplitude and phase measurements will be different for the two

directions. The x stiffness Kxx = 36,000 Lb/in and the y stiffness is Kyy = 140,000 Lb/in.

Fig 27 1 Critical Speed Mode Forst

Multimass Rotor Fig 26 Multimass Rotor in Fluid Film

Bearings

Fig 28 Critical Speed Map For MultiStage

Rotor

Fig 30 4 Tilting Pad Brg CoefficientsFig. 29 5 Tilting Pad Brg Coefficients

13

The x damping for the 5 pad bearing is Cxx = 100 Lb-sec/in and the y damping is Cyy = 210 Lb-sec/in.

Figure 30 shows the corresponding stiffness for a 4 pad bearing configuration with load between pads. In this case,

it is seen that the stiffness and damping coefficients for the 4 pad bearing are symmetric in the x and y directions.

The values for stiffness are Kxx =Kyy = 89,000 Lb/in and the values for damping are Cxx = Cyy = 164 lb-sec/in.

It should be noted that the bearing stiffness and damping coefficients are approximately the average values of the

5 pad bearing configuration. Since the bearing coefficients for the 5 pad bearing are asymmetric, the motion in

general will be elliptical in nature. This means that probes stationed at the horizontal and vertical directions will

observe different values of amplitude and phase. In contrast, the bearing coefficients for the 4 pad bearing are

symmetric. Hence the whirl orbits due to unbalance will be circular in nature. Therefore, the synchronous phase

angle changes with speed will be similar in both the horizontal and vertical directions.

Damped Complex Modes With Tilting 5 Pad Bearing Configuration:

Unbalance Response Amplitude and Phase With Tilting 5 Pad Bearings:

Figure 33 represents the synchronous forced unbalance with the 5 pad bearing configuration. Note that the 1st

critical speed will occur at a speed depending upon which probe one is observing. There is a mild critical speed

observed in the horizontal direction around 2,000 RPM. The major critical speed will be observed in the vertical

direction and this will be at the higher speed of 2,980 RPM. This value corresponds well with the computed damped

1 forward mode of 2,906 CPM. In general, the damped mode frequency will be below or less than the speed ofst

maximum unbalance response. The amplification factor computed by the half power point method is shown to be

3.63 in Fig. 33. Note that the amplification factor calculated from the log decrement as shown in Fig. 31 is given

as 3.75. At the 1 major critical speed at 2,980 RPM, the motion is entirely vertical. There is no x motion!st

Fig. 31 N1f = 2,906 CPM With 5 Pad Brgs,

Log Dec = 0.837 , Ac = 3.75

Fig 32 N2f = 7,250 CPM With 5 Pad Brgs

Log Dec = 3.11 , Ac = 1.01

Fig. 34 Bearing Forces Transmitted With 5 Pad

Bearing Configuration

Fig. 33 Horizontal and Vertical Motion at

St 7 With Tilting 5 Pad Bearings

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In Fig 33, for the horizontal motion, we see that the amplitude continues to increase above the 3,000 RPM speed

range but the phase angle does not. The horizontal phase angle remains relatively constant.

The second damped forward mode as shown in Fig. 32 is a conical mode at a frequency of 7,250 CPM. There does

not appear to be any forced response critical speed in this vicinity. The reason for this is because of the high value

of the log decrement for this mode. As shown in Fig. 32, the log decrement is 3.11. This gives an amplification

factor of 1.01. This means that the mode can not be excited by unbalance. The conical mode is essentially a rigid

body motion with large amplitudes of motion at the bearings. In earlier times, compressor manufacturers were

fearful of operation on top of a critical speed ( some still are). The ability to compute complex damped eigenvalues

was not well established. This lead to many improperly designed compressors with 5 pad bearings with excessively

tight bearing clearances because of misconceptions on 2 critical speed operation and the misconceptions on thend

role of bearing asymmetry on promoting rotor stability.

Damped Complex Modes With Tilting 4 Pad Bearing Configuration:

Figure 35 represents the 1 damped forward mode with 4 pad bearings. The log decrement with the 4 pads isst

superior to the 5 pad design resulting in a lower amplification factor for the 1 mode. Figure 36 represents the 2st nd

forward damped mode. Since the log decrement is over 3.14, there is no apparent 2 critical speed to excite. Itnd

is apparent that the 4 pad bearing has superior damping over the 5 pad bearing configuration and will, in general

,have superior stability characteristics.

Figure 37 represents the amplitude at the rotor center and at the end bearing with the 4 pad bearings. The center

span motion looks like a Jeffcott rotor response with an amplification factor of 2.7. Since there is no second critical

speed at 6,800 RPM, the amplitude of motion at the bearing continues to increase with speed.

Fig. 36 2 Forward Mode at 6,822 CPM Withnd

4 Pad Brgs - Log Dec = 3.232

Fig 35 1 Forward Mode at 2527 CPM Withst

4 Pad Brgs - Log Dec = 1.146 , Ac = 2.74

Fig 37 Amplitude At Center and Brg

Locations With 4 Pad BrgsFig 38 `Bearing Forces Transmitted With

4 Pad Brgs

15

Figure 38 represents the bearing forces transmitted with the 4 pad bearing configuration. After passing through the

1 critical speed, the bearing forces reduce. Above 3,500 RPM, the bearing forces continue to increase. This is duest

to the high value of bearing damping which suppresses the second critical speed. Although we do not encounter a

second critical, we have the undesirable situation of increasing bearing forces with speed. However, the system

behavior with the 4 pad bearings are superior to the response and bearing forces encountered with the 5 pad bearing

configuration. Except for the case of heavy turbines where jacking oil is required for liftoff, there is no advantage to

the use of the 5 pad bearing arrangement for normal compressor design. This is a mistaken design concept that has

persisted for far too long.

Polar Plots Of Rotor Center and Bearing Response With 4 Pad Bearings:

Figure 39 represents the polar plot of the horizontal motion with the 3 planes of unbalance and the rotor supported by

the 4 tilting pad bearing configuration. The response at the rotor center appears to be similar to that of the Jeffcott

rotor. As we move further away from the center, we pick up the effects of the second mode. Figure 40 shows the polar

plot obtained at the first bearing. As the rotor passes through the first critical speed region, the amplitude reaches a

minimum around 3,500 RPM with a maximum phase change of 120 deg. The polar plot then forms a small inner loop

as the amplitude begins to increase without substantial change in the phase angle. Whenever this is observed at the

bearing location, this is an indication that the bearing forces are also increasing.

Fig. 39 Polar Plot of Center Span X Motion

4 Pad Bearings Fig. 40 Polar Plot of Bearing X Motion

4 Pad Bearings

Fig. 42 Rotor Response at 10,000 RPM With

3 Planes Unbalance - 4 Pad Brgs

Fig 41 Rotor Response at 2620 Rpm With

3 Planes of Unbalance- 4 Pad Brgs

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Figure 41 shows the 3 dimensional mode shape at 2620 Rpm. The center plane of unbalance is leading the shaft

deflection by 90 deg. Fig. 42 shows the motion at 10,000 RPM. The center motion is a node point at this speed.

Rotor Response With High Preloaded Tilting 4 Pad Bearings:

The 4 Pad bearing configuration as shown in Fig. 30 has a low preload of 0.3 and a large bearing radial clearance Cb

of 5 mils. This preload and clearance combination leads to a relatively soft bearing. The rotor has a low amplification

factor at the first critical speed and is highly stable from the standpoint of self excited whirl instability with respect

to Alford type cross-coupling forces acting on the impellers. However, since all centrifugal compressor manufacturers

embed their noncontact probes at or near the bearings, there is the impression that the vibrations are excessive and

hence install bearings with higher preloads and tighter bearing clearances.

Figure 43 represents the bearing coefficients for the 4 pad bearing with the bearing preload increased to m = 0.5 and

the radial bearing clearance Cb reduced from 5 to 2 mils. This dramatically increases the bearing stiffness and damping

coefficients by causing the upper pads to be active. Figure 44 represents the resulting 1 forward mode with thest

stiffened bearings. Note, however, the rotor log decrement has been reduced in half.

Influence of Flexibility and Mass on Rotor Response:

In actuality, one can not obtain the full stiffness and damping predicted by a tilting pad bearing program assuming rigid

pivots. The influence of pivot flexibility and foundation flexibility causes a reduction of stiffness, but more

importantly, it can lead to a drastic reduction of damping.

Figure 44 represents the 3 mass rotor mounted on flexible supports. The support flexibility is assumed to be .5e6 Lb/in.

The introduction of support flexibility reduces the first critical speed, but more importantly, the foundation flexibility

can cause a dramatic reduction in the effective damping. Figure 45 represents the amplitude and phase at the first

Fig 44 1 Forward Mode At 4117 CPM With st

4 Pad Brgs, Cb = 2 Mils, LD =.554, Ac=5.67Fig 43 Tilting 4 Pad Brg With Preload m=0.5 and

Bearing Radial Clearance Cb = 2 Mils

Fig. 45 Amplitude and Phase at 1 Bearing stFig 44 3 M ass Rotor With Flexible Supports

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bearing. Note that the computed amplification factor based on the half power method is 7.32. This is in comparison

to the damped eigenvalue analysis of 5.67 based on the bearings in rigid supports. The damping values of 1200 Lb-

sec/in are not obtained in an actual situation.

The introduction of pivot flexibility and foundation flexibility can cause a 50% or higher reduction in the effective

damping in a tilting pad bearing configuration.

Rotor 3 Dimensional Modes Shapes and Bearing Polar Plots With Flexible Supports:

Figure 46 represents the rotor unbalance response at 3,620 RPM which is the critical speed with introduction of the

flexible supports. The center station unbalance at station 5 is leading the center motion by 90 deg. Figure 47 shows

the rotor motion at the higher speed of 12,000 RPM. At this higher speed, the center span motion becomes a node

point and the unbalance at station 5 has no influence on the shaft motion at this elevated speed above the first critical

speed. The bearing motions are out of phase to each other.

Figure 48 represents the polar plot for bearing station 1. The small inner circle represents the passage through the

first critical speed. As the rotor speed exceeds 140% of the first critical speed, which is around 5,000 RPM, the

polar plot moves out at a phase angle of 120 deg and slightly increases in phase. The phase displacement above

Fig. 46 3 D Mode Shape at 3620 RPM

With Flexible Support Kf=.5e6 Lb/In

Fig. 47 3 D Mode Shape at 12,000 RPM

With Flexible Support Kf=.5e6 Lb/In

Fig. 48 Polar Plot of 1 Bearing Motion Fromst

1,000 RPM to 12,000 RPMFig. 49 Polar Plot of 2 Bearing Motion Fromnd

1,000 RPM to 12,000 RPM

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5,000 RPM lags the unbalance at 90 deg at station 3. Since the second critical speed is still well above the maximum

speed of 12,000 RPM, the polar plot moves out radially with little phase change. Figure 49 represents the polar plot

at the second bearing. The 2 unbalances at stations 3 and 7 act as a couple unbalance. Hence to properly balance

the rotor above 5,000 RPM, a two plane balance procedure must be used.

Bifurcated or Split Critical Speeds With Tuned Rotor-Supports:

In the previous case, the introduction of support flexibility causes a reduction of the critical speed and a significant

reduction in the effective bearing damping. When foundation mass is introduced, an interesting effect can occur

if the foundation mass and support natural frequency are close to the rotor critical speed. The foundation interacts

with the rotor to form a tuned resonance condition in which a bifurcated or split critical speed is created.

Figure 50 represents the 1 split critical speed with foundation or support mass of 500 Lb included under the firstst

bearing. The total effective support stiffness is taken as 0.25e6 Lb/in. This creates a foundation resonance around

4100 CPM. Since this frequency is close to the original critical speed with the tilting pad bearings on a rigid

foundation, the critical speed mode is bifurcated into two families. These two frequencies occur above and below

the original critical speed. In Fig. 50, the foundation and 1 bearing motion are in phase. Since the relative motionst

between the between the bearing and support are in phase, the effective damping is reduced. This causes the rotor

critical speed amplification factor to increase to 18 as predicted from the damped eigenvalue analysis. Figure 51

represents the upper branch of the split critical speed mode. In this case, the foundation and bearing are out of phase

and hence the effective damping is increased. For this mode, the amplification factor is predicted to be 8.8.

Therefore the lower branch of the split critical speed may be very difficult to balance because of the reduced

damping. This condition can occur with a soft footing under a machine.

Figure 52 represents the motion at the first bearing and Fig. 53 shows the motion at the rotor quarter span.

Fig. 51 2 Split Critical Speed At 5553 RPMnd

Wf=500 Lb, Kf=.25e6Lb/In , LD=.358, Ac= 8.8

Fig. 50 1 Split Critical Speed At 3087 RPMst

Wf=500 Lb, Kf=.25e6Lb/In , LD=.172, Ac= 18

Fig 52 Amplitude and Phase At 1 Bearingst

Wf = 500 Lb , Kf=.25e6 Lb/InFig 53 Amplitude and Phase At Station 3

Wf = 500 Lb , Kf=.25e6 Lb/In

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Note that in both Figs. 52 and 53, the first branch of the bifurcated critical speed has an amplification factor as computed

by the half power point method of over 29. This condition would make this mode most difficult to balance. This

situation has been encountered with aircraft engine rotor-casing interactions, industrial gas turbines with flexible hot end

supports, and with exciters with damaged grouting. The exciters become almost impossible to balance until the

foundation grouting is repaired. In Fig. 53, note that there is an antiresonance speed at station 3 at 5140 RPM with null

vibrations. Figure 54 shows the motion at the shaft center. Note the high amplification factor of 29.

The high amplification factor encountered at the rotor center

span could lead to seal rubs.Figure 55 represents the forces transmitted at the first bearing with the resonant support

condition under bearing 1.

Discussion and Conclusions:

The knowledge of the behavior of the synchronous unbalanced and forced response phase changes with speed for rotating

machinery can be an important tool in the balancing and diagnostics of many field problems. Unfortunately a large

number of rotors have not been designed to properly incorporate adequate instrumentation to monitor vibration and phase

of motion. This has caused great difficulties with correcting vibration problems encountered with such varied equipment

as the space shuttle engines, aircraft gas turbines, large turbine generators, and centrifugal compressors and large

industrial fans.

The typical phase change of 0 to 180 deg associated with the simple Jeffcott rotor is well understood. The complication

in interpretation occurs when we encounter more complex multimass models with such diverse effects as overhung disks,

shaft bow and mass foundation effects. It is often advisable to have a record of the initial amplitude and phase for

various probes on initial installation of a machine. Significant changes of phase during the operation of the rotor over

time can be indications for maintenance. Without phase readings, single and particularly multi plane balancing is

virtually impossible. Also, by having a proper time mark signal, one can generate Bode and Nyquest plots of the rotor

motion. For aircraft engines, two timing reference signals are required for two spool rotors to separate the high rotor

from the low speed rotor. A great deal of acquired aircraft engine vibration data has often proved to be of limited value

due to the absence of an adequate reference signal for synchronous tracking. This is particularly important when squeeze

film dampers are employed with aircraft engines. When the nonlinear dampers become overloaded, we see the effect by

the change in phase. In a normal rotor, a change in phase with increased unbalance is not pronounced if the rotor is

operating in a linear range. This is not the case with squeeze film dampers as they become highly nonlinear with

increases in unbalance. Synchronous phase monitoring for aircraft engines is extremely important.

A great deal of understanding of the of the relationships between amplitude and phase for multimass rotor system may

be gained by the modeling of the system with the inclusion of such various effects such as skewed disks, misalignment,

shaft bow as well as the inclusion of radial unbalance encountered with rotating equipment. In this presentation, some

of these basic effects are included. Careful inspection of the phase behavior over a speed range for various probes can

significantly improve our understanding of the dynamical behavior of rotating machinery.

Fig 54 Amplitude and Phase at Rotor Center

Wf = 500 Lb , Kf=.25e6 Lb/InFig. 55 Bearing Forces Transmitted With

Resonant Support Condition

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